Forecasting Nonlinear Crude Oil Futures Prices Saeed Moshiri1 Department of Economics University of Manitoba Winnipeg, Manitoba, Canada R3T 5T8
[email protected] Faezeh Foroutan University of Tarbiat Modarres Tehran, Iran
[email protected]
Forecasting Chaotic Futures Crude Oil Prices
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The first author would like to thank the participants for their valuable comments at the 38th annual meetings of the Canadian Economic Association (CEA) at Ryerson University, Toronto, in June 2004. We would also like to thank Laura Brown for reviewing the paper. All possible errors are ours.
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ABSTRACT The movements in oil prices are very complex and therefore, seem to be unpredictable. However, one of the main challenges of the econometric models is to forecast such a seemingly unpredictable economic series. The traditional linear structural models have not been promising when used for forecasting, particularly in the case of complex series such as oil prices. Although linear and nonlinear time series models have performed much better job in forecasting oil prices, there is yet room for an improvement. If the data generating process is nonlinear, applying linear models could result in large forecast errors. Model specification in nonlinear modeling can also be very case dependent and time-consuming. In this paper, we model and forecast daily oil price futures, listed in NYMEX, applying ARIMA and GARCH models, for the period April 1983 – Jan. 2003. We then test for chaos using embedding dimension, BDS, Lyapunov exponent, and neural networks tests. Finally, we set up a nonlinear and flexible ANN model to forecast the series. Since the tests for chaos indicate that the futures oil price follows a chaotic process, the ANN model should make better forecasts. The results of forecasts comparison among different models confirm that. Key words: Crude Oil Price, Nonlinear Dynamic, Chaos, BDS, Laypunov exponent, Neural Networks, Forecasting JEL Classification: C12, C13, C32, C45, C53 1. INTRODUCTION Oil is one of the most important energy resources in the world with wide price swings.
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It has significant effects on the sales and profits of major industries worldwide, and its movements influence capital budgeting plans as well as the value of foreign-denominated asset investments. The crude oil price fluctuations could also bring about economic instability in both oil exporting and oil consuming countries in developed and developing countries. Oil price shocks have often been cited as causing adverse macroeconomic impacts on aggregate output, price, and employment in countries across the world. The Oil price forecasting is, therefore, vital to agents and policy makers. There have been many efforts to exploit models that could explain the changes in the crude oil price and forecast it accurately in spot and exchange trade markets. These models can be grouped into three categories: structural, linear and nonlinear time series models [Bacon (1991), Desbarats (1989)]. The structural models have been able to provide fairly reasonable explanations on the factors underlying the demand and supply movements, but they have not been usually successful in forecasting oil prices (Pindyck, 1999). The linear and nonlinear time series models, such as ARMA and ARCH type models, have been able to do a better job in forecasting oil prices [Abosedra & Laopodis (1997),Morana (2001), Sadorsky (2002)]. However, if the underlying data generating process of the oil prices is nonlinear and chaotic, using the linear or nonlinear parametric ARCH-type models with changing means and variances will not be appealing.
To
forecast a chaotic series, we need a flexible nonlinear and local optimizer model such as an artificial neural network (ANN) model, which is shown to be able to explore the data locally and forecast it more accurately than other competing linear and nonlinear models [Kuan & White (1994), Swanson and White (1997), Moshiri & Brown (2004).] Since the stock market crash of October 1987, researchers have become interested in
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applying chaos theory and have examined new ways to analyze economic and financial time series data. According to the chaos theory, a very complex behavior of economic series, which appear to be random, may be explained by a deterministic nonlinear system. Chaos theory was developed in the fields of mathematics and physics, but has been applied to many other fields such as chemistry, biology, geography, psychology, and recently economics. The existence of the chaotic behavior in economic series, however, is not yet proven. Barnett and Chen (1987) and DeCoster and Mitchell (1991) found evidence of low-dimensional chaos in certain monetary aggregates. However, Brock and Sayers(1988), Frank and Stengos (1988), and Moshiri et al (2002) did not find evidence of chaos in some of the US and the Canadian macroeconomics series including the aggregate consumption series. More studies about the chaos in economics can be found in, Hsieh (1989), Ramsey, Sayers and Rothman (1990), Serletis (1996), Barnett and Serletis (2000), Harrison et al (1999), Gilmore (2001), and Chatrath et al (2002), among the others. Chaos has been also applied to energy markets. Chwee (1998) and Serletis and Gogas (1999) found evidence of chaos in natural gas futures and the North American natural gas liquid markets. Panas and Ninni (2000) used invariant and some non-invariant quantity on daily oil products for the Rotterdam and Mediterranean petroleum market and found strong evidence of chaos in a number of oil products. Adragi and Chatrath (2001) also reported oan evidence of chaos in the oil prices in the futures markets. Very few studies, however, have been carried out to forecast nonlinear deterministic dynamic economic series [Cecen, Erkal (1996).] This paper addresses two questions: First, do linear or nonlinear stochastic models
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explain the crude oil price movements reasonably well or does there exist a deterministic chaotic behavior in the series? Second, if the data generating process of the oil price is chaotic, can the ANN model forecast it more accurately than the traditional econometrics linear and nonlinear models? To address these questions, we will proceed as follows. We will first analyze the price movements of crude oil prices focusing on its stochastic behavior in linear and nonlinear terms. We will then carry out several tests such as correlation dimension, BDS, ANN, and Lyapunov exponent, to investigate if there exists deterministic chaos in the crude oil prices. Finally, we will develop an ANN model to forecast the crude oil prices and compare the results with those obtained from other linear and nonlinear models. Our concluding remarks will end the paper.
2. DATA The futures crude oil price has been fluctuating a lot since the first oil price shock in 1973, when the price was almost quadrupled reaching $12 from $3.40 per barrel. The collapse of the Iranian pro American regime in 1979 and the beginning of the Iraq-Iran war in 1981 both contributed to the oil price more than doubling, reaching $35 per barrel. In 1985, when the world’s largest producer, Saudi Arabia, abandoned its policy of being swing producer in OPEC and increased its production from 2 million barrels per day to 5 million barrels per day, the oil prices plummeted to under $10. Despite moving up and down, the price remained weak until 1999, and started to rise again in the beginning of the new century. Our data consists of about twenty years of daily crude oil futures prices traded at the New York Mercantile (NYMEX) from April 4, 1983 to January 13, 2003, a sum of 5161
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data points. Figure 1 shows the trend and Table 1 the general statistics of the crude oil price in the period. The following general observations can be made from the information provided in the Figure and the Table. [Figure 1 here] 1.
The crude oil price mean in the period is $21.75 with standard deviation $5.53. The
price distribution is not normal, skewed to the right, and fat. 2.
Two major price changes in the oil market have occurred since 1983. The first was
when OPEC raised its production in 1985, and the second when the Persian Gulf war started in 1990. 3.
The oil price fluctuations are not the same in terms of their sizes and duration. This
indicates that a dynamic nonlinear structure may exist in the data suggesting the use of nonlinear models, which are able to capture these irregularities. 4.
The crude oil price does not exhibit a global trend in the period. Despite many
changes, the price has always shown a tendency towards its mean. 5.
According to the ADF test results, we cannot reject the null hypothesis of unit-root
at 1 and 5 percent significant levels. However, the Perron's test result, when the Nov. 1985 shock is taken into account, indicates the process is stationary and, therefore, the data transformation is not necessary. In the following sections, we first estimate the crude oil price by linear and nonlinear models and then use their residuals to test for any unexplained deterministic nonlinearity. [Table 1 here]
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3. LINEAR AND NONLINEAR ESTIMATION MODELS In this section, we estimate the crude oil prices using linear and nonlinear time series models. Applying the Box-Jenkins methodology, ARMA(1,3) is chosen as the best linear model. The details of the results are presented in Table 2. [Table 2 here] The LM serial correlation test result indicates that we cannot reject the null hypothesis of no serial correlation in fitted ARMA model. Although the residuals of the ARMA are uncorrelated, their variances may not be constant over time. Therefore, we carry out another diagnostic test, ARCH-LM test, on the residuals of the ARMA model to see if there exists some nonlinearity not captured by the linear model. The results indicate that the null of constant variance is rejected, suggesting that crude oil price series contain nonlinear dynamics. The GARCH (2, 1) is selected as the best nonlinear model. The details of the results are presented in Table 3. The ARCH-LM test on residuals of this model shows that the heteroscedasticity is removed from the data. [Table 3 here] In the next section, we carry out various tests for deterministic chaos in the crude oil price. We will apply the tests to the residuals of the ARMA and GARCH models specified above to investigate whether any deterministic nonlinearilty remains in the series. .
4. TESTS FOR DETERMINISTIC CHAOS Chaos is a very complex process that looks like random, but it is generated by a
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deterministic and dynamic process [Baumol & Benhabib (1989)]. Since most economic variables are assumed to be random and many theories are formed based on this assumption of randomness, it is imperative to examine if these random variables follow a chaotic process. If so, there will be a change of perspective in the explanation of fluctuations in economic time series, and also in the forecasting modeling. Our preliminary study of the crude oil prices above shows that the linear ARMA and nonlinear GARCH models are able to explain the variations of the price reasonably well resulting in random residuals. Tests for chaos will determine if there still exists a pattern, a complex and deterministic one, in the residuals. If so, there will be room for improvement in forecasting by applying flexible nonlinear models. There are a variety of tests available for detecting chaos. Since the nulls and the alternatives in the existing tests are not the same, one cannot decide as to which test has higher power [Barnett & Serletis (2000)]. This makes the distinction between deterministic and stochastic process by applying a single statistical test difficult. Therefore, in order to avoid misleading results and conclusions, we carry out a group of tests available within a nonlinear framework. The tests applied are correlation dimension, BDS, ANN, and Lyapunov exponent tests. The correlation dimension is a non-statistical test, which uses integral correlation to test for chaos. The BDS tests for iid versus a general nonlinearity in the series, and the ANN tests for linearity versus a general nonlinearity. The BDS and ANN tests do not provide clear evidence for the presence of chaos, even when the null of iid or linearity has been rejected. The Lyapunov exponent test, however, can be considered as a more direct test for chaos, since it is based on one of the main characteristics of the chaotic series, namely, the extreme sensitivity to the initial condition.
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4.1. The Correlatoin Dimension Test Correlation dimension (cd) , also called fractal dimension, is a measure to determine the degree of complexity of a time series, and therefore, can be used for testing for chaos. For a chaotic process, cd assumes some non-integer values and also converges to a saturation limit. Grossberger and Procaccia (1983), calculate the cd using Integral correlation as follows.
Let Xt ,t = 1...,T , be a time series. It can be converted to m-dimensional as X t ( x t , x t ,..., x t ( m 1) )
Xt is an m-tuple vector which can be considered as a point in the m-dimensional space. There will be N (=T-m+1) m-tuple vectors in m-dimensional space. m is the embedding dimension.
is a time delay used in constructing the m-histories of the series and is
chosen in such a way as to avoid too high a correlation between the elements of an m-tuple. It is usually recommended to set this lag equal to the first zero-crossing of the autocorrelation function. We follow that here. Integral correlation is defined as
C m ( )
N 1 H ( X j X i ) N ( N 1) i , j 1
where ε is a predefined small value, and H ( X j X i ) Heaviside step function defined as
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H=
{
1 , if H ( X j X i ) 0 0 , if H ( X j X i ) 0
Integral correlation computes the number of m-tuple vectors in phase space whose distances do not exceed ε. The correlation dimension is then defined as
log C m D Lim log m
Given ε, in a deterministic chaotic series, as the embedding dimension increases, m
the number of pairs within ε decline, and, therefore, D reaches a finite saturation limit, whereas in a stochastic series, it continues to rise. The results of the correlation dimension calculated for the crude oil price series and the residuals of the ARMA and GARCH models are presented in Table 4. [Table 4 here] There are two concerns about the calculation of cd for a time series. First, as Harrison et al (1999) note, the presence of noise may result in an increase in cd as m increases. Therefore, using cd as a test for chaos in a noisy data may be misleading. Second, as Scheinkman and Lebaron (1989) point out, while an increase in m would not affect the estimate of cd after a certain point for infinite series, it would for finite data. This is particularly important in economic and financial series where the number of observations is very limited compared with empirical sciences. Therefore, the test results based on cd in finite data cannot be conclusive. To address the noise problem, we apply two noise reduction techniques, namely HP and Wavelet, to the crude oil price and recalculate the cd. The results indicate no significant changes in the pattern of cd even
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after the removal of noise from the data in the case of the residuals, but a convergence in the original data. The second problem does not apply here, as our sample size is rather large. However, to examine the crude oil price series more carefully, we will carry out more rigorous tests below.
4.2 BDS Test The BDS test; named after Brock, Dechert, and Scheinkman (1988), is a statistical version of the correlation dimension test for randomness or “whiteness” against the alternative general dependence in a series. The BDS test can also be used to produce indirect evidence about non-linearity in the data. If an ARMA process extracts a linear structure from the data, then the BDS test can be applied to the residuals to determine whether they are white noise. If the null of white noise is rejected, then there exists a general dependence in the residuals, which may be due to the neglected non-linearity in the estimation process. In this case, further investigation is needed to narrow down the alternative and determine the causes for the failure of the linear process. The BDS test is based on the concept of correlation integral used by Grassberger and Procaccia (1983) in tests for chaos and non-linearity. Where the observations ( xt) are identically and independently distributed, the true correlation integral for dimension m (Cm,T(ε)) is related to the correlation integral for dimension 1 by the relation Cm,T(ε) = m
[C1,T(ε)] , for all m and ε. Brock et al. (1988) showed that if the points xt are iid, the m
standardized difference between Cm,T(ε) and [C1,T(ε)] is asymptotically normally
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distributed. That is, 1/2
m
Wm,T(ε) = T [Cm,T(ε) - (C1,T(ε)) ]/σm,T(ε) ~ N(0,1), where T is the sample size, Cm,T(ε) is the correlation integral, and σm,T(ε) / T
1/2
is the
standard deviation of Cm. The BDS test is able to pick up deviations from linearity in mean, and is also sensitive to series having autoregressive conditional heteroscedasticity (Lee et al, 1993). In this study, we will first apply the BDS test to the residual of the ARMA model estimated in section 3. The residuals of the model should be in principle linear independent, and therefore any dependency found in the residuals must be due to ignored nonlinearity. However the absence of chaos will be implied if it is demonstrated that the nonlinear structure arises from a known non-deterministic system (Brock et al, 1993). In this case, the BDS test is run on the standardized residuals from an ARCH-type model, and if the null hypothesis has been accepted, it can be concluded that the ARCH-type process is able to explain any non-linear structure in the data. As the results presented in Table 5 show, we reject the null of iid for the residuals of ARMA model except for ε = 0.5 in different dimensions. This reveals that it may be possible for residuals to remain nonlinear, assuming that all linearity from the data has been already eliminated. The results of the BDS test applied to the residuals of the GARCH model also indicate that the null of iid is rejected, suggesting that other nonlinear structures may exist in the data, or it follows a chaotic process.
4.3 The Artificial Neural Network Test
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Artificial Neural Network (hereafter, ANN) models are non-linear input-output models with certain special features such as mass parallelism and non-linear processing of inputs, which are also found in biological neural networks. In general form, the ANN output vector produced by a model or network with one output unit can be written as q
y = F[β0 + Σ j=1 G(x γj) βj] ≡ f(x, θ) where y is the network’s final output, F and G are (usually non-linear) transformation functions, x = [1, x1, ..., xr] is a matrix of r input vectors (including the intercept term), β= [β0, β1 , …β
q]
is a vector of weights for the q intermediate transformations by the function
G, γ = [γ1, , γq] is a matrix of q weight vectors, each vector relating the r input variables to one of the q intermediate totals, and θ = (γ1, ,
γq, βj) is a general term for the matrix
containing both sets of weights. F and G can take any functional form, but the non-linear -αx
sigmoid function [y = 1/(1+e )] is a popular one, particularly for G (Kuan & White, 1994.) In order to use the ANN modeling approach to test for linearity, we first add an extra, direct connection between inputs x and output y, and then assume that the output transformation function (F) is linear. With those two changes and after adding an error term (ε), the output function becomes as follows. y = β0 + x δ + Σ
q
j=1G(x
γj) β j + ε
where δ is a (1xr) vector of connection weights between inputs (x) and output (y), and other symbols are as in the original equation. If the process underlying the time series x is linear, the non-linear component of the equation, i.e. the third term, would vanish, leaving a linear regression model with the coefficients β0 and the vector δ. Therefore, the neural network test for non-linearity uses equation above testing the null hypothesis β j = 0, j = 1,
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.. ,q. Similar to the BDS test, if we apply a linear filter, such as ARMA, to a time series, the residuals can then be used to determine whether there exists a non-linearity in the series. If series is linear, the residuals obtained from a linear process should not be correlated with ARMA process and any function of the history. Thus, the null hypothesis can be expressed as E(etGt)=0, where et are the residuals from a linear regression of y on x and Gt is the vector of q intermediate totals for date t. To avoid the problem of collinearity between x and the components of G, the principal components of G which are not correlated with x can be used instead of G. In this case, the following test statistics can be used 2
2
TR → χ (q), 2
where T is the number of observations and R is the squared multiple correlation coefficient from a standard linear regression of e on x and principal components of G which are not correlated with x (Kuan and White, 1994). The results of the ANN test applied to the crude oil prices are presented in Table 6. According to the test results, the hypothesis of linearity is rejected at 5 percent significance level. [Table 6 here] The results of the BDS and the ANN tests are consistent with each other, but neither is a direct test for nonlinearity and chaos; the rejection of the null hypothesis could result from either a nonlinear stochastic or a nonlinear deterministic system. A more direct test for chaos is to compute the Lyapunov exponent, which is presented in the following section.
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4.4 Lyapunov Exponents (LE) Test One of the most important characteristics of chaos is the sensitivity of the process to the initial condition. Lyapunov exponents measure the sensitivity of the system to changes in initial condition or determine the stability of the periodic orbits [Bask (1996)]. In particular, LE reveals the existence of deterministic chaos in time series by measuring the degree of divergence of nearby trajectories in the phase space. A positive value of the largest LE is a sign of deterministic chaotic behavior [Wolf and Swift, 1985).] LE calculation is based on dimensions of the phase space. There are as many LE as the dimensions. The LE algorithm is as follows. First, consider a matrix X of dimension (T-m+1) × m, where T is the length of original time series and m is the embedding dimension. Second, choose any two arbitrary row vectors betweem which the Euclidian distance is less than a pre-specified small value, e, as follows:
r0 X i X j e
If the process is chaotic, at the next n time-step-ahead, the two vectors Xi+n and Xj+n will be diverged. Third, define dn , as the ratio of distance between pairs of X in n time-step-ahead and initial time, as follows,
dn
X i n X j n Xi X j
and calculate it for different n values. If dn exceeds one, conclude that process is chaotic, since by increasing the time step, close points in m-dimensional state space will be
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divergent. LE can be obtained as follows.
LE1 (m, n) lim
1 log dn (m; i, j) T n n
A positive value of LE implies that the points in m-dimensional space, in an attractor of a nonlinear process, will be diverged when increasing the time step. LE can also be calculated using an alternative definition of dn as the ratio of rn to rn−1, rather than r0, as follows.
LE2 (m, n) Lim
1 log dn (m, i, j) T n n
where
d n
X in X j n X i ( n 1) X j ( n 1
As the estimation results in Table 7 indicate, the estimated Lyapunov exponent is positive for all selected embedding dimensions, suggesting the presence of chaos in the process generating the crude oil price series. These results support the results obtained by the previous tests carried out earlier.
[Table 7 here]
5. FORECASTING CRUDE OIL PRICE Traditional econometrics models do not have a good record in forecasting. Although time series models, in general, do a better job, flexible and nonparametric models, such as artificial neural networks (ANN), are superior in forecasting nonlinear
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and complex series [McMenamin and Monfrote (1998), Swanson and White (1997), Zhang et al (2001), and Moshiri and Brown (2004)]. Since the crude oil price, as shown in the previous section, follows a chaotic process, we expect that the ANN model would generate more accurate forecasts. In this section, we will develop a feedforward multilayer perceptron (MLP) model to forecast the crude oil price and compare the results with those from linear and nonlinear, i.e. ARMA and GARCH, models applied in section 3. The ANN model consists of a lagged price as input, one hidden layer with five operating units (neuron), and one output. Each input pattern is composed of a moving window of fixed length along the series. The transfer function in the hidden layer is tansig and in the output layer is identical linear function. The learning algorithm in the network is gradient descent. In order to evaluate the forecast results, the data set is divided into two parts: estimation (training), and forecasting (test). The estimation period
is
1983:4:4-2000:6:25 and the forecasting period 2000:6:26-2003:1:13, leaving us with 4461 and 700 observations for estimation and forecasting, respectively. The forecasting method is dynamic, where the estimated prices, not the actual one, are used for lagged prices when forecasting next period. To compare the forecasting results obtained by different models, three forecasting error functions are calculated. They are Mean Absolute Error (MAE), Mean Square Error (MSE), and Root Mean Square Error (RMSE). We use EViews 4 for estimation and forecasting the crude oil price by ARMA and GARCH models, and MATLAB6 by ANN model. The forecasting results by the three models are presented in Table 8. [Table 8 here] As the results in Table 8 show, the ANN model has been able to produce lower
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forecasting errors by all three measures. To test for statistical significance in forecasting errors generated by alternative models, we use Diebold-Mariano (1995) test. The test statistic is defined as follows. s
1
Where d
d t 1
, and is its asymptotic variance estimated as:
Tf
Tf
d
t
k 1 1 t 0 2 i i 1
where k is forecasting horizon, i the ith estimation of auto covariance and 0 the estimation for variance. d is the difference between squared errors generated by two alternative models. Under the null of equal forecasting error functions, s is asymptotically normally distributed with mean zero and standard deviation 1. The Diebold – Mario statistic for testing statistical difference between ARMA and ANN forecasting errors is -8.11 and GARCH and ANN Models 7.13, suggesting that the differences between forecasting errors are statistically significant.
6. CONCLUSION Forecasting the crude oil price is still one of the big challenges facing economists and econometricians. For the past two decades, many models have been developed to identify the data generating process of the oil prices and to produce more accurate forecasts. Fortunately, recent advances in the computing technology have enabled
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econometricians to develop and to work with nonlinear and complex models with intensive computations more easily. These models seem to be more suitable for estimation and, in particular, for forecasting very complex series such as the crude oil prices. In this paper, we examined the statistical features of the crude oil price series, using the daily data in NYMEX futures markets in 1983-2003. The traditional view is that the price variations can be explained by a trend, history, stochastic factors, or a combination of them. However, if the data generating process of the crude oil prices is chaotic, the traditional modeling for estimation and forecasting would not be appealing. To shed more lights on the underlying data generating process of the crude oil prices, we carried out various tests for deterministic chaos. The tests included correlation dimension, BDS, ANN, and Lypunov Exponent. Given the restrictions of the tests, and the fact that each test is developed based on certain characteristics of the chaotic process, the individual results obtained form the tests cannot be conclusive. Therefore, applying all available tests and comparing the results would allow us to get more robust outcomes. All our test results suggest that the daily crude oil prices reported in NYMEX in 1983-2003 follow a nonlinear dynamic and deterministic process. If we know the economic structure underlying the chaotic process, we would be able to forecast the series very accurately, at least in the short run. Unfortunately, the tests for chaos are not able to point to a specific economic structure. They only tell us about the possibility of existing a nonlinear stochastic or nonlinear deterministic process underlying the data. Forecasting a chaotic series, without the knowledge of the specification of its structure, therefore, would require a flexible, nonlinear, and local optimizer model such as ANN
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model. We developed a feedforward multilayer ANN model to estimate and forecast the crude oil prices, and compared the results with those obtained by the standard ARMA and GARCH models. The ANN model outperformed the linear and nonlinear models. It may be worthwhile to apply some other nonlinear time series models; such as Threshold AR, Exponential AR, or Generalized AR; which may produce the same results as the ANN. The problem with using this approach is that there are many different nonlinear specifications from which one should be chosen [Granger (1998).] As Zhang (2001) notes, while these models can be useful for a particular problem and data, they do not have general appeal for other applications and one particular nonlinear specification may not be general enough to capture all nonlinearity in the data. The main advantage of the models such as ANN is that they are universal approximator, that is, they are able to learn (estimate) all kinds of data and forecast them with reasonable accuracy.
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business 62 No.3 Serletis A. (1996), Is there chaos in economic time series?. Canadian Journal of Economics, April. Swanson N., White H. (1997), A model selection approach to real-time macroeconomic forecasting using linear models and artificial neural networks. The Review of Economics and Statistics 79(4): 540-550. Wolf, A. Swift, J., Swinney, H., Vastando, J. (1985), Determining Lyapunov Exponent from a Time Series. Physica 16 D 285-317 Zhang, G. Peter. Patuwo, B. Eddy. Hu, Michael Y. (2001), A Simulation Study of Artificial Neural Networks for nonlinear Time Series Forecasting. Computer and Operations Research 28 381-396.
24
25
45
Series1
Sep -2000
40
Oct-90 40.42 35
Nov-85 31.72
Sep- 2001 27.7
30
25
20
15
jul -90 16.94
10
Dec -98 11.23 5
2002/04/04
2001/04/04
2000/04/04
1999/04/04
1998/04/04
1997/04/04
1996/04/04
1995/04/04
1994/04/04
1993/04/04
1992/04/04
1991/04/04
1990/04/04
1989/04/04
1988/04/04
1987/04/04
1986/04/04
1985/04/04
1984/04/04
1983/04/04
0
Figure 1. Time plot of daily futures crude oil price X
i
Figure 2. Two nearby trajectories separating as time goes by.
26
Table 1- Statistical prosperities of crude oil Mean
Std. Dev
Skewness
Kurtosis
Jarqe-Bera
ADF
Perron’s Test
21.75
5.53
0.45
2.37
262.867(0.00)
-2.83*
-4
In ADF test, a constant term is included in the estimation along with one to three lags; the critical values reported for Augmented Dickey-Fuller test are -3.43, -2.86 and -2.56 at 1, 5 and 10 percent levels of significance, respectively. The Perron's test is the estimation of price lag coefficient with λ =0.1, and the critical value is -3.68.
Table 2-ARMA Estimates for the crude oil prices (1983-2003) X t 0 1 X t 1 1Z t 2 2 Z t 3 D1 t 1
0
1
1
1
2
20.82 (12.79)
-1.46(-3.86)
0.99 (492.46)
-0.06 (-1.82)
-0.12 (-2.36)
Serial correlation LM-test 0.5 ( 0.60)
ARCH-LM Test 59.99 (0.00)
Std. Dev
Skewness
Kurtosis
AIC
BIC
SSE
0.48
-2.72
63.22
1.38
1.38
958.94
Values in parentheses in the first row are t-statistics, and in the second row p-values.
Table 3-GARCH Estimate for the crude oil prices (1983-2003) log( t2 ) 0 log( t21 ) 1 log( t2 2 )
-0.142 (-7.80)
0
t 1 t t 1
1
0.338 (5.67)
-0.165 (-2.87)
ARCH-LM Test 0.19 (0.66)
0.991 (275) AIC 0.72
SIC 0.73
SSE 971.17
Values in parentheses in the first row are t-statistics, and in the second row p-values.
27
Table 4- The correlation dimension for futures crude oil price (1983-2003) Series
Embedding dimension
CD - Crude oil price CD – ARMA residuals CD – GARCH residuals
4
8
12
16
20
0.436
0.472
0.502
0.529
0.553
0.412
0.668
0.866
1.025
1.186
0.402
0.662
0.845
1.009
1.155
Table 5- The BDS test results on the residuals of the ARAM and GARCH models for the crude oil price (1983-2003) series
m
ε/σ
0.5
1
1.5
2
2
1.23
1.8
1.7
1.24
3
1.53
3.5
3.66
2.85
1.35
4.6
5.44
4.48
5
1.02
5.2
6.99
6.1
6
0.73
5.5
8.33
7.75
2
1.27
1.92
1.69
1.18
3
1.60
3.63
3.62
2.73
4
1.43
4.79
5.42
4.31
5
1.10
5.45
7.00
5.89
6
0.80
5.74
8.4
7.45
4 ARMA residuals
GARCH residuals
m is the embedding dimension. ε/σ is equal to 0.5, 1, 1.5 and 2 times the standard deviation divided by spread of the residual series. The critical values are 1.645, 1.96, and 2.575 for the 10%, 5% and 1% significant levels, respectively.
Table 6 -The results of the ANN test for linearity in the crude oil prices (1983-2003) R2 0.019
TR2 Test Statistic 12.65
Degrees of freedom
Critical value (5%)
5
11.07 (0.05) 16.75 (0.005)
28
Table 7-Results of Lyapunov exponent test for chaos in crude oil prices (1983-2003) m 2
ε/σ
LE1 0.3784
LE2 0.0013
0.4585
0.0011
1.0750
3.3935e-004
5
1.1623
6.1229e-004
6
1.1593
5.4231e-004
2
0.3840
-2.7372e-005
3
0.3689
0.0011
0.3591
0.0011
5
0.4223
0.0012
6
0.3807
0.0012
2
0.3344
8.7432e-004
3
0.4782
6.2734e-004
3 4
4
4
0.5
1
0.3870
0.0013
5
1.5
0.3441
0.0010
6
0.3689
0.0010
2
0.4212
7.0586e-004
0.4372
8.2168e-004
0.4563
6.1595e-004
5
0.5034
7.0368e-004
6
0.3080
0.0011
3 4
2
m is embedding dimension. ε/σ is equal to 0.5, 1, 1.5 and 2 times the standard deviation/spread of the series.
Table 8 -The forecasting results by the linear and nonlinear models for the crude oil price (2000-2003) Model
MSE
RMSE
MAE
ARMA
29.27
5.41
4.81
GARCH
15.25
3.90
2.90
ANN
8.14
2.85
2.04
MSE: Mean squared error, MAE: Mean absolute error, RMSE: Root Mean Square Error
29
